# How do you find the inflection points of #f(x)= 12x^5+45x^4-80x^3+6#?

Maximum: x=-4

Inflection: x=0

Minimum: x=1

This is a challenging problem for a quintic equation in general, but in this case, the quintic has some low order terms missing that greatly aid in our understanding.

This is a quartic equation, so we can factorize it. The complete "solution by radicals" of the quartic is known, but it is lengthy, intricate, and excruciating to carry out.

Therefore, the function's maximum is -4 and its minimum is +1. We must take another derivative in order to classify 0.

As a result, 0 is an inflection point.

Plot it twice; once zoomed in and once out: graph{12x^5+45x^4-80x^3+6 [-5, 2, -20, 20]} graph{12x^5+45x^4-80x^3+6 [-5, 2, -1000, 5000]}. We want to sanity check our answer by comparing it to the function's graph, but the maximum at -4 is off the scale compared to the other points of interest.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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